Apologies again for the lack of posts. I'm currently working on a rather long writeup, detailing C* algebras, the completion of the convolution algebras, the presentation of and the arithmetic subalgebra of the bost connes system, and it's interaction with the class field theory of Q. Perhaps its a bit too ambitious for a single blog post. I'm trying to pin down exactly what I'm confused on. I've not been able to prove the presentation of the system, or even understand what I read on it.
In any case, I'm giving a talk at the UCL Undergrad Maths Colloquium on my project, and thought I'd post my notes here. You can download them from the colloquium from this link.